Peel the mango and cut the flesh away from the pit. Cut the flesh into chunks and put them in a blender with the chopped pineapple, sugar, water, lime juice, rum and salt. Squeeze the mango pit hard over the blender to extract as much of the pulp and juice as possible. Puree the mixture until smooth. Taste, then add more lime juice or rum if desired.
Chill the mixture thoroughly.
Freeze it in your ice cream maker according to manufacturer's instructions.